Would it be harder to pronounce or would it hinder the flow of speech? Would it make it harder to recognize where a word stops and where the next starts?
Although not for every word, the vast majority of words in Italian that start with a consonant end with a vowel, except loanwords like computer.
That's because during the passage from Latin to Italian, words ending with -um and -us lost their finals and replaced those with vowels, such as Rosam -> Rosa or Lupum -> Lupo.
Also Japanese works in a similar fashion, since the sounds are syllabic (CV and a CCV), except the only isolated consonant N.
I don't know how this would translate to a language where every word starts with a consonant sound and ends with a vowel sound, although I don't think there would be a problem. If you're creating this language and you see that in large discourses it tends to crate problems, you could place some "device" to help diminish those consequences.
Irrespective of the words beginning with a C and ending with a V, most languages have different means to help the listener notice the word boundary.
For example, some languages have only monosyllabic words, so a syllable = a word.
Others have vowel harmony, when the vowels inside a word have a common feature while the neighboring words are likely to have this feature different, so when the feature changes, a new word begins.
Stress in all of its kinds is also a great helper, e.g. in Czech all the words are stressed on the first syllable, so a stressed syllable signals the beginning of a new word. In Russian, different syllables can be stressed, but the unstressed ones are reduced and pronounced more quietly, so the loudness of speech changes in a saw-like graph, peaks being the stressed vowels, and the valleys the word boundaries.
There are two general phonological tendencies that are related: the requirement for every syllable to begin with a consonant (e.g. Arabic), and the requirement for every syllable to end with a vowel (e.g. Maori). Taken together, these requirements give you languages which have just CV syllables (Senufo). However: your question was stated in terms of words, not syllables, and as such, that describes a language type that I don't think is known to exist, namely one that doesn't care about consonant clusters and vowel clusters, but requires words to begin with C and end with V.
Pronunciation wouldn't be a problem, since syllables at word edges would be a subset of syllables pronounced elsewhere. For the same reason, parsing words would not be a problem, and could be mildly easier (since any sequence C+C or V+V would not have a word break, thus the second segment would not be that of a word-initial morpheme).
"Would it make it harder to recognize where a word stops and where the next starts?"
This is the problem of segmentation of an utterance into morphemes, the difference between "experts' exchange" and "expert sex change". Languages have a couple strategies to solve this, which apply whether or not a language has CV syllable structure.
One strategy is to adopt predictable stress patterns. In Toki Pona, for instance, the first syllable of a word is always stressed, and the modifier in the phrase ends up carrying the phrase's primary stress. For example the phrase jan pona, which means "friend", behaves as a compound noun (/ˌjanˈpona/ which may realize in running speech as [ˌjɐmˈb̥ɔnɐ]) with pona "good" as its root. Stress in Spanish is not as completely predictable (and it's slightly less CV anyway), but the majority of words that end in a vowel are stressed on the penult (second to last) syllable. In a language with long vowels or geminate consonants, stress may fall a given number of morae, or length units smaller than a syllable, from the end of the word, as in Latin and Hawaiian.
Another is to relax the stress constraints, but bias the lexicon against roots that contain the same sounds as common inflections or particles. For example, if two phrases supa yuza and su payuza invite confusion, people start rephrasing their utterances to avoid them. This is also fairly common and plausible, but it does result in occasional puns, as shown in the "Mondegreen" and "Pen Island" page on the literary analysis wiki All The Tropes.